1. SIGNALS AND SYSTEMS USING MATLAB
Chapter 1 — Continuous–time Signals
L. F. Chaparro and A. Akan
2. Classification of time–dependent signals
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• Predictability: random or deterministic
• Variation of time and amplitude: continuous-time, discrete–
time, or digital
• Energy/power: finite or infinite energy/power
• Repetitive behavior: periodic or aperiodic
• Symmetry with respect to time origin: even or odd
• Support: Finite or infinite support (outside support signal is
always zero)
3. Analog to digital and digital to analog conversion
t
−∆/
2
∆
Ts
2T
s
3T
s
4Ts
x(nTs)
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• Analog to digital converter (ADC or A/D converter): converts analog
signals into digital signals
• Digital to analog converter (DAC or D/A converter): converts digital
to analog signals
level
x(t)
∆
∆/ 2
Discretization in time and in amplitude of analog signal using sampling period Ts and
quantization level ∆. In time, samples are taken at uniform times {nTs }, and in
amplitude the range of amplitudes is divided into a finite number of levels so that
each sample value is approximated by one of them
4. 0 2 4 8 10
−1
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
6
t (sec)
v(t)
1.45
0.5
0.4
0.3
0.2
0.1
0
−0.1
−0.2
−0.3
−0.4
−0.5
s
v(t),
v(nT
s
)
1.45 1.5 1.55
−0.5
0
0.5
v(nT
s
),
v
q
(nT
s
)
1.5 1.55
1.45
t, nT
−0.5
0
0.5
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1.5
1.55
nTs
e(nT
s
)
Segment of voice signal on top is sampled and quantized. Bottom left: voice
segment (continuous line) and the sampled signal (vertical samples) using a
sampling period Ts = 0.001 sec. Bottom-right: sampled and quantized signal at
the top, and quantization error, difference between the sampled and the
quantized signals, at the bottom.
6. Basic signal operations
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Given signals x (t), y (t), constants α and , and function w(t)
τ :
• Signal addition/subtraction: x(t) + y (t), x (t) − y (t)
• Constant multiplication: αx (t)
• Time shifting
• x (t − τ ) is x (t) delayed by τ
• x (t + τ ) is x (t) advanced by τ
• Time scaling x(αt)
• α = −1, x (−t) reversed in time or reflected
• α > 1, x (αt) is x (t) compressed
• α < 1, x (αt) is x (t) expanded
• Time windowing x(t)w(t), w(t) window
• Integration
8. Delayed, advanced and reflected signals
x(t
)
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x(t −
τ)
x(t +
τ)
x(−t
)
τ
τ
t t
t t
(a) (b)
(c) (d)
Continuous-time signal (a), and its delayed (b), advanced (c), and reflected (d) versions.
10. Even and odd signals
x(t)
even:
odd :
x (t) = x (−t)
x (t) = −x (−t)
• Even and odd decomposition: For any signal y (t)
y (t) = ye (t) + yo (t)
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ye (t) = 0.5 [y (t) + y (−t)]
yo (t) = 0.5 [y (t) − y (−t)]
even component
odd component
Example: x (t) = cos(2πt + θ), − ∞ < t < ∞
even x (t) = x (−t) → cos(2πt + θ) = cos(−2πt + θ) = cos(2πt − θ) θ
= −θ, or θ = 0, π
odd x (t) = −x (−t) → cos(2πt + θ) = − cos(−2πt + θ) = cos(−2πt + θ
± π)
= cos(2πt − θ π)
∓
θ = −θ π, or θ = π/2
∓ ∓
11. Example:
x (t) =
2 cos(4t) t
> 0
0 otherwise
not even or odd, its even and odd components are
−1 −0.5 0 0.5 1
2
1
0
−1
−2
x(t)
−1 −0.5 0.5 1
1
0.5
0
−0.5
−1
0
t
x
(t)
e
−1 −0.5 0.5 1
1
0.5
0
−0.5
−1
0
t
11 /
x
(t)
o
If signal is 2 at t = 0
1
x (t) =
2 cos(4t) t ≥ 0
0 otherwise
the odd component is same as before, and the even component is 2 at t = 0 and same as
before otherwise
13. Finite–energy and finite-power signals
• x (t) is finite–energy, or square integrable, if Ex < ∞
• x (t) is finite–power if Px < ∞
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14. Power of periodic signal
x (t) period of fundamental period T0 is
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21
15. Basic
signals
• Complex exponential
x (t) = Aeat
= |A|ej θ
e(r +j Ω0)t
rt
= |A|e [cos(Ω0t + θ) + j
sin(Ω0t + θ)]
− ∞ < t <
∞
−2 2
4
3
2
1
0
t
e
−0.5
t
−2 2
4
3
2
1
0
t
e
0.5
t
−2 2
−4
−2
0
2
4
0
t
e
−0.5
t
cos(2
t)
−2 2
−4
−2
0
2
4
0
t
e
0.5
t
cos(2
t)
Analog exponentials: decaying exponential (top left), growing exponential (top right),
modulated exponential decaying and growing (bottom left and right).
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16. • Sinusoi
d
A cos(Ω0t + θ) = A sin(Ω0t + θ + π/2) − ∞ < t < ∞
Modulation systems
A(t) cos(Ω(t)t + θ(t))
• Amplitude modulation or AM: A(t) changes according to the message,
frequency and phase constant,
• Frequency modulation or FM: Ω(t) changes according to the message, amplitude
and phase constant,
• Phase modulation or PM: θ(t) changes according to the message, amplitude and
frequency constant
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21
17. • Unit-impulse signal
p∆(t)
1/ ∆
t t
t
t
1
0.5
−∆/2 ∆/2
δ(t)
(1
)
u∆ (t)
−∆/2 ∆/2
u(t)
1
Unit-impulse δ(t) and unit–step u(t) as ∆ → 0 in pulse p∆(t) and its integral u∆(t).
17 /
20. ρ(t
)
dρ(t
) dt
20 /
1
1
1
2 2
3
3
0 0
t t
(−2
)
(−2
)
(2)
1
The number in () is area of the corresponding delta signal and it indicates the jump at
the particular discontinuity, positive when increasing and negative when decreasing
(1)
21. Generic representation of signals
f (t
)
δ(t − t0) f(t0)δ(t −
t0)
t0
t t0 t t
(f (t0)
)
• Generic representation
t t t
∆
x(t
)
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x∆ (t)
+
∆
∆ 2∆
x(0
)
x(∆
)
+
![Even and odd signals
x(t)
even:
odd :
x (t) = x (−t)
x (t) = −x (−t)
• Even and odd decomposition: For any signal y (t)
y (t) = ye (t) + yo (t)
10 /
ye (t) = 0.5 [y (t) + y (−t)]
yo (t) = 0.5 [y (t) − y (−t)]
even component
odd component
Example: x (t) = cos(2πt + θ), − ∞ < t < ∞
even x (t) = x (−t) → cos(2πt + θ) = cos(−2πt + θ) = cos(2πt − θ) θ
= −θ, or θ = 0, π
odd x (t) = −x (−t) → cos(2πt + θ) = − cos(−2πt + θ) = cos(−2πt + θ
± π)
= cos(2πt − θ π)
∓
θ = −θ π, or θ = π/2
∓ ∓](https://image.slidesharecdn.com/chapter01-250713130649-0645f517/85/signals-and-systems-using-matlab-ch1-pptx-10-320.jpg)
![Basic
signals
• Complex exponential
x (t) = Aeat
= |A|ej θ
e(r +j Ω0)t
rt
= |A|e [cos(Ω0t + θ) + j
sin(Ω0t + θ)]
− ∞ < t <
∞
−2 2
4
3
2
1
0
t
e
−0.5
t
−2 2
4
3
2
1
0
t
e
0.5
t
−2 2
−4
−2
0
2
4
0
t
e
−0.5
t
cos(2
t)
−2 2
−4
−2
0
2
4
0
t
e
0.5
t
cos(2
t)
Analog exponentials: decaying exponential (top left), growing exponential (top right),
modulated exponential decaying and growing (bottom left and right).
15 /](https://image.slidesharecdn.com/chapter01-250713130649-0645f517/85/signals-and-systems-using-matlab-ch1-pptx-15-320.jpg)
